It takes a bit of searching to find Sergey Brin’s office at the Googleplex. Tucked away in a corner of Building #43 on this sprawling campus near the southern tip of San Francisco Bay, past rows of colorfully decorated cubicles and dorm-like meeting spaces, Office 211 has a nondescript exterior and sits far from the public eye. Although it takes several twists and turns to get there, his office is not protected—as you would expect for the cofounder of a $150-billion company—by a Russian nesting doll’s worth of doors and gatekeepers. Sergey, 33, shares the space with his Google cofounder, fellow Stanford Ph.D. dropout and billionaire pal, 34-year-old Larry Page, an arrangement that began eight years ago in the company’s first humble headquarters in a Menlo Park, California, garage. Since then, Google has grown from just...

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